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Random arcs on the circle

Published online by Cambridge University Press:  14 July 2016

Andrew F. Siegel
Andrew F. Siegel*
Affiliation:
University of Wisconsin-Madison
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Abstract

Place n arcs of equal lengths randomly on the circumference of a circle, and let C denote the proportion covered. The moments of C (moments of coverage) are found by solving a recursive integral equation, and a formula is derived for the cumulative distribution function. The asymptotic distribution of C for large n is explored, and is shown to be related to the exponential distribution.

Keywords

GEOMETRICAL PROBABILITYRANDOM ARCSRANDOM SETSCOVERAGEMOMENTS OF COVERAGEINTEGRAL EQUATIONVACANCY
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 4 , December 1978 , pp. 774 - 789
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Ailam, G. (1966) Moments of coverage and coverage spaces. J. Appl. Prob. 3, 550–555.Google Scholar
[2] Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
[3] Cooke, P. J. (1972) Sequential coverage in geometrical probability. Stanford University Technical Report No. 198.Google Scholar
[4] Darling, D. A. (1953) On a class of problems related to the random division of an interval. Ann. Math. Statist. 24, 239–253.Google Scholar
[5] Domb, C. (1947) The problem of random intervals on a line. Proc. Camb. Phil. Soc. 43, 329–341.Google Scholar
[6] Feller, W. (1968), (1971) An Introduction to Probability Theory and its Applications , Vol. 1, 3rd edn; Vol. 2, 2nd edn. Wiley, New York.Google Scholar
[7] Flatto, L. and Konheim, A. G. (1962) The random division of an interval and the random covering of a circle. SIAM Rev. 4, 211–222.Google Scholar
[8] Robbins, H. E. (1944) On the measure of a random set. Ann. Math. Statist. 15, 70–74.Google Scholar
[9] Shepp, L. A. (1972) Covering the circle with random arcs. Israel J. Math. 11, 328–345.Google Scholar
[10] Stevens, W. L. (1939) Solution to a geometrical problem in probability. Ann. Eugenics 9, 315–320.Google Scholar
[11] Votaw, D. F. Jr (1946) The probability distribution of the measure of a random linear set. Ann. Math. Statist. 17, 240–244.Google Scholar